Determine whether ∑ 1 / n1 + 1/n converges

Test the following series for convergence:

$\sum_{n=1}^{\infty} \frac{1}{n^{1 + \frac{1}{n}}}.$

The series diverges.

Proof. First, we write

$\frac{1}{n^{1+\frac{1}{n}}} = \frac{1}{n \cdot n^{\frac{1}{n}}}.$

Then, we know $n < 2^n$ for all $n \geq 1$ . (We can deduce this from the Bernoulli inequality, $(1+x)^n \geq 1 + nx$ with $x = 1$ . We proved the Bernoulli inequality in this exercise, Section I.4.10, Exercise #14.) Therefore, $n^{\frac{1}{n}} < 2$ and we have